Single-stage optimization example notebook

[IssraAli]

Created a single-stage optimization example notebook.

[review-notebook-app]

Check out this pull request on 

See visual diffs & provide feedback on Jupyter Notebooks.

---

Powered by ReviewNB

[github-actions]

|             benchmark_name             |         dt(%)          |         dt(s)          |        t_new(s)        |        t_old(s)        | 
| -------------------------------------- | ---------------------- | ---------------------- | ---------------------- | ---------------------- |
 test_build_transform_fft_lowres         |     -5.45 +/- 3.81     | -2.95e-02 +/- 2.06e-02 |  5.12e-01 +/- 9.0e-03  |  5.42e-01 +/- 1.9e-02  |
 test_equilibrium_init_medres            |     -8.23 +/- 3.43     | -3.64e-01 +/- 1.51e-01 |  4.05e+00 +/- 1.0e-01  |  4.42e+00 +/- 1.1e-01  |
 test_equilibrium_init_highres           |     -0.51 +/- 6.06     | -2.85e-02 +/- 3.41e-01 |  5.59e+00 +/- 2.9e-01  |  5.62e+00 +/- 1.8e-01  |
 test_objective_compile_dshape_current   |     +1.85 +/- 3.28     | +7.11e-02 +/- 1.26e-01 |  3.92e+00 +/- 9.9e-02  |  3.85e+00 +/- 7.8e-02  |
 test_objective_compute_dshape_current   |     +4.84 +/- 3.36     | +1.68e-04 +/- 1.17e-04 |  3.64e-03 +/- 1.1e-04  |  3.47e-03 +/- 4.8e-05  |
 test_objective_jac_dshape_current       |     +0.45 +/- 4.95     | +1.75e-04 +/- 1.93e-03 |  3.92e-02 +/- 1.4e-03  |  3.91e-02 +/- 1.3e-03  |
 test_perturb_2                          |     +0.26 +/- 4.97     | +4.50e-02 +/- 8.64e-01 |  1.74e+01 +/- 5.8e-01  |  1.74e+01 +/- 6.4e-01  |
 test_proximal_freeb_jac                 |     -0.66 +/- 2.37     | -4.97e-02 +/- 1.78e-01 |  7.45e+00 +/- 6.9e-02  |  7.50e+00 +/- 1.6e-01  |
 test_solve_fixed_iter                   |     +1.50 +/- 59.20    | +7.06e-02 +/- 2.79e+00 |  4.78e+00 +/- 2.1e+00  |  4.71e+00 +/- 1.9e+00  |
 test_build_transform_fft_midres         |     -1.42 +/- 3.38     | -8.67e-03 +/- 2.06e-02 |  6.03e-01 +/- 1.6e-02  |  6.12e-01 +/- 1.3e-02  |
 test_build_transform_fft_highres        |     -1.75 +/- 4.02     | -1.77e-02 +/- 4.07e-02 |  9.96e-01 +/- 3.8e-02  |  1.01e+00 +/- 1.6e-02  |
 test_equilibrium_init_lowres            |     -2.00 +/- 5.11     | -7.76e-02 +/- 1.98e-01 |  3.81e+00 +/- 4.5e-02  |  3.89e+00 +/- 1.9e-01  |
 test_objective_compile_atf              |     -1.86 +/- 3.76     | -1.47e-01 +/- 2.98e-01 |  7.78e+00 +/- 2.5e-01  |  7.92e+00 +/- 1.7e-01  |
 test_objective_compute_atf              |     +0.31 +/- 3.36     | +3.13e-05 +/- 3.43e-04 |  1.03e-02 +/- 3.2e-04  |  1.02e-02 +/- 1.2e-04  |
 test_objective_jac_atf                  |     -0.78 +/- 4.02     | -1.49e-02 +/- 7.63e-02 |  1.88e+00 +/- 4.7e-02  |  1.90e+00 +/- 6.0e-02  |
 test_perturb_1                          |     -1.30 +/- 1.28     | -1.62e-01 +/- 1.59e-01 |  1.23e+01 +/- 6.8e-02  |  1.24e+01 +/- 1.4e-01  |
 test_proximal_jac_atf                   |     -0.49 +/- 0.78     | -3.88e-02 +/- 6.23e-02 |  7.93e+00 +/- 4.9e-02  |  7.97e+00 +/- 3.8e-02  |
 test_proximal_freeb_compute             |     +0.07 +/- 0.98     | +1.21e-04 +/- 1.77e-03 |  1.81e-01 +/- 1.4e-03  |  1.81e-01 +/- 1.1e-03  |

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 95.33%. Comparing base (2ebdf6a) to head (c4a09cd).

Additional details and impacted files

@@           Coverage Diff           @@
##           master    #1236   +/-   ##
=======================================
  Coverage   95.33%   95.33%           
=======================================
  Files          90       90           
  Lines       22702    22702           
=======================================
  Hits        21644    21644           
  Misses       1058     1058           

see 1 file with indirect coverage changes

[ddudt]

Noting that this will have to wait for #1222 and possibly also #1205

[dpanici]

Noting that this will have to wait for #1222 and possibly also #1205

IDT it needs to wait for #1205 if it is vacuum, though yea we should wait for both of these if we want a finite beta case also in this

[IssraAli]

I wanted to mention, certain combinations of hyperparameters gave very bad outputs, see below for an example. I didn't really systematically document this, but I can do that if you guys are interested (sorry for the bad pic):

The optimizer also converged to designs where the coils were all together in the centre of the torus all tangled up. Perhaps it would be worth it to create a constraint function that ensures that the coils are within a certain "winding volume" padding the torus to prevent these kinds of outputs.

I generally got much more normal looking coils with the coil-related losses weighted on the order of 100x higher than the QS loss. However, there seems to be quite a lot of field error.